Solution To Numerical Analysis Problem (4)

 As astute math nerds would have quickly discovered in addressing this problem, there is no proper solution - by which I mean the convergence closes and yields more exact results than Newton's method. Indeed, any initial inputs used will be seen to quickly diverge.  First consider the ideal circumstance by which to apply the secant method:

Here the targeted root lies on the secant line and on the abscissa (x-axis) so the application becomes a 'gimme'.   But this is not the case with the given problem, i.e.

For which no constructed secant line will get us what we want.  Consider drawing the only secant line available given that the curve has a local maximum at  - 1/ Ö3  and a local minimum at 1/ Ö3  which define a region wherein we would not expect good results.  So on drawing the secant suggested we get:

But the best we get is the first iteration choosing end points at  x_o=  3.0  and  x₁=  1.3 (instead of 2.11 which is off the secant line) but is essentially near the root sought. Working - we come up with (from Mathcad):

And we see that we obtain a result (1.3739) that is within 0.0492 of the actual root value, i.e. obtained via 5 iterations with the Newton method (1.3247).  However, using the same secant method  in the next iteration we only obtain 2.0914 which is further from the desired result.  So the secant method does not improve on what the Newton method achieved even in its 1st approximation (1.325).  What we see happen is the results diverge from the actual root.